Positive Entire Solutions Of Singular Non-linear Harmonic Equations(Systems)

In the paper we dicuss the existence of positive entire solutions of singular non-linear harmonic equations and their systems.Our work includes the following three parts.Part 1. We consider semi-linear harmonic equations of the formΔu + f(x,u,â–½u) u^-λ=0 , x∈R^N (N≥3) withλ∈(0,1), and give some sufficient conditions for these equations to have a non-radial positive entire solution.Part 2. We consider the systems of p- harmonic equations of the formΔ(|Δⁿu_j |p^j-2 (Δⁿu_j)) = f_j (| x |, (u₁, u₂), (|â–½u₁|, |â–½u₂|)) u_j^-βj ,x∈RN (N≥2),j=1,2 and obtain some existence theorems, in plane and in RN (N≥3) respctively, for these systems to have infinitely many radially positive entire solutions with the prescribed asymptotic behaviors at infinity. Our existence theorems are concerned not only sufficient conditions, but also necessary conditions.Part 3. We consider the systems of p- laplace equations of the divergence form and give some sufficient conditions for these systems to have infinitely many radially positive entire solutions which are asymptotic to positive constant multiples of |x| ^{(pj-N)/(pj-1)} for p_j >N or log| x | for pj=N as | x |→∞.Comparing with the work of many authors, the equations (systems) we deal with are more general and complex, since the equations here are all singular and the functions f are also depending on the gradientsâ–½u, and in Parts 2 and 3, the higher order equations are studied. Therefore, the applications may be extended and, at the same time, more difficults must be faced with. Owing to the combinational employment of super-subsolutions and fixed point theorem as well as potential theory, we obtain the existence theorems for each parts and forthur some related results.